Heat-transfer coefficients in laminar flow

Annuli Approximate heat-transfer coefficients for laminar flow in annuh may be predicted by the equation of Chen, Hawkins, and Sol-berg [Tron.s. Am. Soc. Mech. Eng., 68, 99 (1946)] ... 

When two or more phases are present, it is rarely possible to design a reactor on a strictly first-principles basis. Rather than starting with the mass, energy, and momentum transport equations, as was done for the laminar flow systems in Chapter 8, we tend to use simplified flow models with empirical correlations for mass transfer coefficients and interfacial areas. The approach is conceptually similar to that used for friction factors and heat transfer coefficients in turbulent flow systems. It usually provides an adequate basis for design and scaleup, although extra care must be taken that the correlations are appropriate. 

Knudsen and Katz [54] have shown that it is valid for Re Pr- djl >10. Equation 19.21 cannot be used for long tubes, since it would yield zero heat transfer coefficient. Sarti et al. have employed a different Equation 19.16 to estimate the heat transfer coefficient for laminar flow in circular tubes (shown in Table 19.1). 

Recently, Phattaranawik et al. [48] have used several equations to estimate the heat transfer coefficient in laminar and turbulent flow regimes. They found that Equation 19.22 is the most suitable for laminar flow, while the Dittus-Boelter equation was most suitable for turbulent conditions. 

The pioneering conclusion drawn by Tuckerman and Pease in 1982 [1] that the heat transfer coefficient for laminar flow through microchannels may be greater than that for turbulent flow, accelerated research in this area. Many experimental [2-6], numerical [7-10], and analytical [11-14] studies have been performed, with some focusing on the effects of roughness [15-21] and temperature-variable thermophysical properties of the fluid [22-26]. 

As seen in the previous section, flow is considered to be laminar when Re < 2300 and turbulent when Re > 104. Transition flow occurs in the range of 2300 < Re < 104. Few correlations or formulas for computing the friction factor and heat transfer coefficient in transition flow are available. In this section, the formula developed by Bhatti and Shah [45] is presented to compute the friction factor. It follows ... 

Correlations are available for predicting pressiffe drops and convective heat transfer coefficients for laminar flow inside and outside of ducts, tubes, and pipes for pipes with longitudinal and peripheral fins for condensation and boiling and for several different geometries used in compact heat exchangers. No attempt is made to discuss or summarize these correlations here. They are presented by Hewitt (1992). 

Laminar Flow Although heat-transfer coefficients for laminar flow are considerably smaller than for turbulent flow, it is sometimes necessary to accept lower heat transfer in order to reduce pumping costs. The heat-flow mechanism in purely laminar flow is conduction. The rate of heat flow between the walls of the conduit and the fluid flowing in it can be obtained analytically. But to obtain a solution it is necessary to know or assume the velocity distribution in the conduit. In fully developed laminar flow without heat transfer, the velocity distribution at any cross section has the shape of a parabola. The velocity profile in laminar flow usually becomes fully established much more rapidly than the temperature profile. Heat-transfer equations based on the assumption of a parabolic velocity distribution will therefore not introduce serious errors for viscous fluids flowing in long ducts, if they are modified to account for effects caused by the variation of the viscosity due to the temperature gradient. The equation below can be used to predict heat transfer in laminar flow. 

The recommended procedure is to calculate the heat transfer coefficient using both mechanisms and select the higher value as the effective heat transfer coefficient (h). For baffled condensers, the vapor shear effects vary for each typical baffle section. The condenser should be calculated in increments with the average vapor velocity (Vy) for each increment used to calculate vapor shear heat transfer coefficients. When the heat transfer coefficients for laminar flow and for vapor shear are nearly equal, the effective heat transfer coefficient (h) is increased above the higher of the two values. The table below permits the increase to be approximated ... 

Buller and Kilburn (1981) performed experiments determining the heat transfer coefficients for laminar flow forced air cooling for integrated circuit packages mounted on printed wiring boards (thus for conditions differing from that of a flat plate), and correlated he with the air speed through use of the Colburn J factor, a dimensionless number, in the form of... 

Performance problems related to maldistribution also exist in cooling applications, especially where viscosity increases as temperature is lowered. At worst, case equipment could become inoperable, due to plugging of all but the center of the flow channel. This condition can be eliminated by the use of static mixer internals, discussed later. Heat transfer coefficients for laminar flow in empty pipe are correlated by... 

Chapter 7 deals with the practical problems. It contains the results of the general hydrodynamical and thermal characteristics corresponding to laminar flows in micro-channels of different geometry. The overall correlations for drag and heat transfer coefficients in micro-channels at single- and two-phase flows, as well as data on physical properties of selected working fluids are presented. The correlation for boiling heat transfer is also considered. 

Select the appropriate heat-transfer coefficient equation. Heat-transfer coefficients for fluids flowing inside helical coils can be calculated with modifications of the equations for straight tubes. The equations presented in Example 7.18 should be multiplied by the factor 1 + 3.5/1, /D,. where Di is the inside diameter and Dc is the diameter of the helix or coil. In addition, for laminar flow, the term (Dc/Dj)1/6 should be substituted for the term (L/Zl)1 3. The Reynolds number required for turbulent flow is 2100[1 + I2(/1,//1C)I/2. ... 

For laminar and turbulent flows, we need appropriate correlation equations for the friction coefficient, heat transfer coefficient, and mass transfer coefficient. For laminar flow in the ranges of 5 X 106 > Re > iO3, and Pr and Sc > 0.5, we have the following relations for the coefficients ... 

At Reynolds numbers greater than about 30, it is observed that waves form at the liquid-vapor interface although the flow in liquid film remains laminar. I he flow in this case is said to be wavy laminar. The waves at the liquid-vapor interface tend to increase heat transfer. But the waves also complicate the analysis and make it very difficult to obtain analytical solutions. Therefore, we have to rely on experimental studies. The increase in heat transfer due to the wave effect is, on average, about 20 percent, but it can exceed 50 percent. The exact amount of enhancement depends on the Reynolds number. Rased on his experimental studies, Kutateladze (1963) recommended the following relation for the average heat transfer coefficient in wavy laminar condensate flow for p p, and 30 < Re < 1800,... 

The unusual behavior of Nu decreasing with increasing Re in the laminar regime in microchannels may alter the status of thermal development and hence the conventional thermal entry length, since the variation of the heat transfer coefficient along the flow is a variation of the boundary condition. 

Heat transfer coefficients in thermally fully developed, laminar flow... 

Recently, Hedrick [12] developed a new correlation for the heat transfer coefficient in the transition region between laminar and turbulent flow. The equations for determining the inside film coefficient and based on the outside tube diameter, hj , are ... 

Heat transfer to a laminar flow in an annulus is complicated by the fact that both the velocity and thermal profiles are simultaneously developing near the entrance and, often, over the length of the heated channel. Natural convection may also be a factor. It is usually conservative (i.e., predicted heat-transfer coefficients are lower than those experienced) to use equations for the fully developed flow. 

---